Optimal. Leaf size=14 \[ (d-2 e) \log (x+2)+e x \]
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Rubi [A] time = 0.0239435, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {1586, 43} \[ (d-2 e) \log (x+2)+e x \]
Antiderivative was successfully verified.
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Rule 1586
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (2-x-2 x^2+x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x}{2+x} \, dx\\ &=\int \left (e+\frac{d-2 e}{2+x}\right ) \, dx\\ &=e x+(d-2 e) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0044482, size = 16, normalized size = 1.14 \[ (d-2 e) \log (x+2)+e (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 1.3 \begin{align*} ex+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948895, size = 19, normalized size = 1.36 \begin{align*} e x +{\left (d - 2 \, e\right )} \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73338, size = 38, normalized size = 2.71 \begin{align*} e x +{\left (d - 2 \, e\right )} \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.263152, size = 12, normalized size = 0.86 \begin{align*} e x + \left (d - 2 e\right ) \log{\left (x + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08089, size = 23, normalized size = 1.64 \begin{align*} x e +{\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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